jaj•a•person's notes

Tag: state/tidy

See Tag Glossary.

986 items with this tag.

(Extra)natural transformation
∗-representation of the complex group ring
2 central extension of an elementary abelian 2-group
3b1b's eigenvalue trick
A bijection is open iff it is closed
A continuous bijection from compact to Hausdorff is a homeomorphism
A covering is injective on the fundamental group
A covering is regular iff its deck transformation group acts transitively on fibres
A deck transformation maps sheets to sheets
A Dedekind domain admits UFI
A Dedekind domain is a CDR
A Dedekind domain is a UFD iff its ideal class group is trivial
A Dedekind domain with finitely many prime ideals is a UFD
A field contains modular arithmetic or the rationals
A finite integral domain is a field
A group homomorphism induces a subgroup homomorphism
A map is a homeomorphism iff it is bijective, continuous, and open
A probable cladistic tree of IE
A quadric is singular iff its matrix is singular away from 2
A ring contains the integers or modular arithmetic
A seminorm induces a normed quotient
A topological space is discrete iff every point is its own connected component
𝑎𝑏 and 𝑏𝑎 have the same order
Abelian category
Abelian group
Abelian groups as Z-modules
Abelian Lie algebra
Abelian representation
Abelianization
Ablaut
Absolute norm of an ideal of the ring of integers of a number field
Absolutely homogenous function
Abstract projective plane
Abstract projective space
Action functional
Acyclic quiver
Adjoining a root to a field
Adjoining the degree derivation
Adjoint Lie algebra representation
Affine Lie algebra
Alexander subbasis theorem
Alexandrov extension
Algebra homomorphism
Algebra of Laurent polynomials
Algebraic closure
Algebraic element
Algebraic integer
Algebraic interior of a field extension
Algebraically closed field
All elements of a finite-dimensional unital associative algebra are algebraic
All primes are irreducible in an integral domain
Almost everywhere
Alternating character
Alternating group
Alternating iff anticommutative away from 2
Amalgamated free product
Ampère's circuital law
Ampère's law for magnets
An algebraic element is invertible iff its minimal polynomial has a nonzero constant term
Annihilator
Annoying index
Associate elements
Associate representation
Associated Legendre function
Atlas
Autistic category
Automorphism
Automorphism of a field extension
Away from 2
Axiom of Choice
Axiom of Extensionality
Axiom of Foundation
Axiom of Infinity
Axiom of Intersection
Axiom of Purity
Axiom of Union
Axiom Schema of Replacement
Axiomatic set theory
Balanced product
Ball space
Bases for a number field
Bernoulli differential equations
Bernoulli trial
Beta distribution
Bianchi identities
Bicartesian category
Bilinear form
Binomial distribution
Biproduct
Borel set
Bound on the automorphism group of a finite simple extension
Bounded operator
Bounded set
Braided monoidal category
Bundle map
Bundle section
Canonical projections are open
Canonical tensors over a nondegenerate quadratic space
Cardinality
Cartesian category
Categories are equivalent iff they have isomorphic skeleta
Category
Category of categories
Category of coverings
Category of field extensions
Category of fields
Category of G-spaces
Category of graded Lie algebras
Category of graded modules
Category of group representations
Category of metric spaces and isometries
Category of monoidal categories
Category of paths
Category of quadratic Lie algebras
Category of quadratic spaces
Category of vector bundles
Category-vector space analogy
Cauchy sequence
Cauchy-Euler differential equations
Cauchy's order theorem
Cayley's theorem
Central group extension
Central ideal
Central limits theorem
Centralizer in a group
Centralizer in a Lie algebra
Centre of a group
Centre of a Lie algebra
Centre of a rng
Centre of the general linear group
Centre of the group ring
Chain map
Change of inner product
Character table
Characteristic
Characteristic conjugacy class of a path-connected covering
Characteristic function (probability)
Characteristic subgroup of a covering
Characters of a finite symmetric group are real
Charge continuity equation
Chebyshev's inequality
Chernoff's inequality
Chinese remainder theorem
Chinese remainder theorem for rings
Choice function
Christoffel symbols
Circle endomorphisms are homotopic iff they are of equal degree
Class function
Classifying DEs
Closed category
Closed category of graded vector spaces
Closed subsets of a compact space are compact
Coarseness and fineness of topologies
Cocartesian category
Code
Commutative diagram
Commutative ring
Commutator
Commutator ideal
Commutator subgroup
Compact sets in a metric space are bounded
Compact space
Compact subsets of a Hausdorff space are closed
Complement Axiom for classes
Complement subspace
Completeness and cocompleteness
Complex quadrance
Complexification of a real Lie algebra
Compositions only of algebraic extensions are algebraic
Condition for a quotient commutative ring to be a field
Condition for a quotient commutative ring to be an integral domain
Conditional distribution
Conditional probability
Conditional proof
Conditions for uniqueness of the limit
Conductor in electrostatic equilibrium
Cones and cocones
Congruence relation
Conjugacy classes of a symmetric group are determined by cycle structure
Conjugate subgroups
Conjugate transpose
Conjugation by an element
Connected subgroup
Connected subspaces of the real line are intervals
Connectedness
Connectedness is transitive
Connex category
Connexion disagreement tensor
Conservative vector field
Constant map
Containment-division ring
Continuity
Continuous group action
Continuous path
Continuous random variable
Contractible space
Contraction map
Contraction map theorem
Contravariant form on a triangular module
Contravariant functor
Conventions of 0th-order logic in these notes
Convergence in distribution
Convergence in probability
Convex function
Convex subset
Convolution of group characters
Convolution of matrix representations
Coördinate chart
Coproduct topology
Core IE thematic subjunctive
Correspondence between normal subgroups and congruence relations
Correspondence between quadratic forms and alternating bilinear forms at 2
Correspondence between quadratic forms and symmetric bilinear forms away from 2
Correspondence between regular coverings and orbit spaces of their deck transformation groups
Cosets are either identical or disjoint
Coulomb gauge
Coulomb's law
Countable Axiom of Choice
Cover set
Covering
Criterion for 𝑎ⁱ = 𝑎ʲ in a group
Crystallographic restriction theorem
Cyclic central extension of a free abelian group
Cyclic subgroup
Deck transformation group of a regular covering as quotient
Decomposable module
Deformation retraction
Degree of a circle endomorphism
Degree operator
Degree operator on formal sums over a vector space
Derivation on an algebra
Derivation subalgebra
Determinant
Diagonalization
Diagonalization of a quadratic form away from 2
Diameter of a set
Diëlectric
Differentiability
Differential pullback along a diffeomorphism
Differential system
Dihedral group
Dimension of a vector space
Direct product of groups
Direct product of Lie algebras
Direct sum of chain complexes
Direct sum of graded vector spaces
Direct sum of linear maps
Direct sum of modules
Direct sum of representations
Discrete category
Discrete random variable
Discrete subgroup
Discriminant of a number field
Discriminant of an algebraic integer
Disjoint category
Distribution of a differentiable injective random function
Distribution of microstates at equilibrium
Divergence
Divisible code
Division algebra
Division algebra with only algebraic elements over an algebraically closed field
Dixmier's lemma
Domain Axiom for classes
Dominating measure
Effective group action
Eigenvectors, eigenvalues, and eigenspaces
Electric dipole moment
Electric displacement
Electric field
Electric potential of a polarized material
Electromagnetic energy density
Electromagnetic momentum density
Elementary abelian group
Elementary row operation
Elementhood Relation Class Axiom
Embedding
Embedding an algebraic extension into an algebraically closed field
Empty category
Emptyset Axiom
Endomorphism
Endomorphism ring
Energy
Energy of an ideal gas
Engel's theorem
Enriched functor
Epimorphism
Equivalence of categories
Equivalence of codes
Equivalence of coverings criterion
Equivalence of group representations
Equivalence of irreps on left ideals criterion
Equivalence of norms
Equivalence relation
Essential supremum and infimum
Euclid's lemma
Euler totient function
Even lattice
Every arc-transitive digraph is an orbital digraph
Every fibre of a local injection is discrete
Every finite complex representation of a compact group is equivalent to a unitary representation
Every irrep of GLₙ(ℂ) is an irrep of U(n) and SU(n)
Every vector space has a basis
Exact differential equation
Exact functor on abelian categories
Exact sequence
Examples of monoidal categories
Excess kurtosis
Existence and uniqueness theorem for IVPs
Expectation
Exponential of a derivation on a Lie algebra
Extended code
Extended Euclid's algorithm
Extension field as a unital associative algebra
Exterior algebra
Extraspecial p-group
Extreme Value Theorem
Faithful functor
Faithful module
Faraday's law of induction
Fermat's little theorem
Ferromagnetism
Fibre bundle
Fibre product and coproduct
Fibre product is the equalizer of a product
Field
Field extension
Field homomorphisms are injective
Filtered algebra
Finite extension of a Galois field
Finite field extension
Finite group character values
Finite projective plane
Finite projective space
Finitely generated field extension
Finitely generated module over a module-finite R-ring
First countability axiom
First law for isobaric processes
First law of thermodynamics
Five lemma
Fixed field of an automorphism group
Fixed order subgroup of an abelian group
FLM code types I and II
Flux
Formal logarithm
Formal sums over a Lie algebra
Fréchet space
Free category
Free group
Free group action
Free module
Free product of groups
Freshman's dream
Frobenius endomorphism
Fully faithful functor
Functional
Functor category
Functors encode invariants of isomorphism classes
Fundamental group respects products
Fundamental theorem for line integrals
Fundamental theorem of arithmetic
Fundamental theorem of cyclic groups
Fundamental unit of a real quadratic field
Galois field
Galois geometry
Gauß-Jordan elimination
Gauß's law
Gauß's law for diëlectrics
Gauß's law for magnetic flux
Gaußian elimination
Gaußian integers
Gaußian pillbox
GCD is a linear combination
General random variable
General structure of a mathematics area
Generalized eigenvector
Generalized Hamming weight
Generalized projection operator of a representation
Germanic Strong Conjugation
Gibbs free energy
Glossary of categories
Graded algebra
Graded derivation
Graded dimension
Graded Lie algebra
Graded module
Graded ring
Graded structure
Graded subalgebra
Graded submodule
Graded vector space
Graded vector subspace
Gram matrix
Graph automorphism
Graph set
Graphical neighbours
Graphs as quivers
Green's function
Group action
Group action orbit
Group action orbital
Group character
Group class function
Group commutator
Group epimorphism
Group extension
Group homomorphism
Group isomorphism theorems
Group monomorphism
Group of prime order
Group of roots of unity
Group of units
Group order
Group presentation
Group representation
Group ring
Hamming code
Hamming distance
Hausdorff-compact space
Heat
Heat of transformation
Heat reservoir
Heine-Borel theorem
Heine-Cantor theorem
Heisenberg algebra
Heisenberg module
Helmholtz free energy
Helmholtz theorem
Hermitian operator
Hilbert's basis theorem
Hölder conjugate
Hölder's inequality
Hom-functor
Homeomorphism
Homogenous coördinates
Homogenous first-order differential equation
Homogenous function
Homogenous linear ODE with constant coëfficients
Homogenous system of linear equations
Homography
Homomorphism of comonoid objects
Homomorphism of graded vector spaces
Homomorphism of monoid objects
Homomorphism of semigroup objects
Homotopy of paths
Hook length
Hook length formula
Ideal
Ideal class group
Ideal class group of a number field
Ideal gas
Ideal gas law
Ideal of the complex group ring
Ideals of a Dedekind domain need at most two generators
Idempotent primitivity criterion
Identity as a natural transformation
Identity natural transformation
Image and preïmage
Imaginary quadratic field
Immersion and submersion
Incidence relation
Incompressible vector field
Independence of random variables
Induced module
Initial and terminal objects
Inner group automorphism
Inner Lie algebra derivation
Inner product space
Integral domain
Intermediate field extension
International Keyboard Cheatsheet
- public
- state/tidy
Invariant bilinear form on a Lie algebra
Invariant subspace
Invariant subspaces of ∗-representations and unitary representations coïncide
Irreducible character as function of an idempotent
Irreducible element
Irreducible operators applied to an irreducible orthonormal basis transform in the product representation
Irreducible orthonormal basis
Irrep
Irreps collectively distinguish group elements
Irreps of abelian groups are 1-dimensional
Isomorphism
Isomorphism between the complex group ring and direct sum of matrix algebras on carriers of irreducible representations
Isomorphism of categories
Iverson bracket
Japanese allophones
Joint cumulative distribution function
Joint probability mass function
K-monoid
Kernel of a group homomorphism
Kernel of a Lie algebra homomorphism
Kernel of an algebra homomorphism
Kernels and cokernels
Kronecker delta
Krull dimension
Ladder operator
Lagrange multiplier
Lagrange's theorem
Lattice subgroup
Law of large numbers
Lebesgue number
Lebesgue sequence space
Lebesgue space
Left and right continuity
Legendre symbol
Levi-Civita connexion
Liar paradox
Lie algebra
Lie algebra automorphism
Lie algebra extension
Lie algebra homomorphism
Lie algebra ideal
Lie algebra isomorphism
Lie algebra isomorphism theorems
Lie algebra of nilpotent endomorphisms
Lie algebra representation
Lie algebras MOC
Lie subalgebra
Lift of a map to a covering space
Limit point
Limits and colimits
Lindeman's option
Linear (in)dependence
Linear code
Linear commutator
Linear diëlectric
Linear differential system
Linear endomorphism
Linear epimorphism
Linear equations MOC
Linear equivalence of codes
Linear isomorphism
Linear kernel
Linear magnet
Linear map
Linear quiver
Local (path) connectedness
Local finite measure of a compact set is finite
Local injection
Local Lagrangian
Local topological group
Locally path-connected, connected covering morphism is a covering
Locally small category
Location-scale transformation
Log-normal distribution
Loop algebra
Lorentz force law
Lower bound on the dimension of the field of rational functions
Macroscopic state variables of an ideal gas
Magma object
Magmoid
Magmoid homomorphism
Magnet
Magnetic dipole moment
Magnetic field
Magnetic potential of a magnetized material
Main theorem of connectedness
Main theorem of coverings
Map of Content
Markov's inequality
Maschke's theorem
Material set theory
Matrix determinant is a homomorphism
Matrix of a system of linear equations
Maximal ideal
Maximal ideal iff prime ideal in a PID
Maxwell's equations
Maxwell's equations in materials
Mean value theorem
Measurable function
Measure space
Measure theoretic pigeonhole principle
Memoryless random variable
Method of undetermined coëfficients
Method of variation of parameters
Metric completion
Metric topology
Metrizable implies first-countable
Metrizable implies Hausdorff
Minimum of independent exponentially distributed random variables
Minkowski embedding
Minor
Module
Module over a category ring
Module over a group
Module over a Lie algebra
Module over a unital associative algebra
Module-finite R-monoid
Moment-generating function
Monogenic field
Monoid ring
Monoidal functor
Monoidal natural transformation
Monomial transformation
Monomorphism
Morphism
Morphism of field extensions
Multifunctor
Multilinear map
Multiplicative group of integers modulo 𝑛
Multipole expansion of the electrostatic potential
Multiset
Multivariate normal distribution
Multivariate random variable
Natural Heisenberg algebras
Natural isomorphism
Natural transformation
Neighbourhood
Neighbourhood basis
Neighbourhood system
Nested neighbourhood basis
Nilpotent Lie algebra
No group is the union of two proper subgroups
Noetherian ring
Normal closure
Normal quadratic subspace
Normal subgroup
Normalizer in a group
Normalizer in a Lie algebra
Normed vector space
Null-homotopic map
Number field
Number of elements of each order in a cyclic group
Number of elements of order 𝑑 in a finite group
Number of points in a finite projective plane
Number of subspaces of a Galois geometry
Numeric partition
Objects as functors
OIcel a-stems
OIcel declension of adjectives
OIcel feminine root nouns
OIcel i-stems
OIcel j-insertion
OIcel masculine root nouns
OIcel n-stems
OIcel nd-stems
OIcel ō-stems
OIcel progressive consonant assimilation
OIcel r-stems
OIcel u-umlaut
OIcel v-insertion
Oidification
Open ball
Operator norm
Opposite category
Orbit space
Orbit space of a properly discontinuous effective group action
Orbit space of a properly discontinuous group action
Orbit-stabilizer theorem
Order of powers of a group element
Ordinal category
Orthochronous Lorentz group
Orthogonal code
Orthogonal complement
Orthogonality by a quadric
Orthonormal basis
Orthonormal set
Orthonormality of irreducible characters
Orthonormality of irreps
p-group
𝑝-series
Parallel transport
Partial derivative as a local affine connexion
Path connectedness
Path Lagrangian
Path traversal lemma
Perfect code
Perfect magnetic dipole
Permutation group
Phase change
Physics MOC
PIE *e-laryngeal colouring
PIE *s insertion
PIE aspect
PIE case
PIE conjugation
PIE laryngeal deletion
PIE mood
PIE syllabification of sonorants
PIE thematic vowel
PIE velar labialisation
PIE verb endings
PIE verb stem formation
PIE voice
PIE vowel lengthening by contraction
Pigeonhole principle
PKM MOC
Plücker embedding
Poisson algebra
Polynomial ring
Poset
Posetal category
Posets as categories
Powerset Axiom
Poynting vector
Poynting's theorem
Preïmage theorem
Preorder
Pressure
Prime ideal
Prime ideals are invertible in a Dedekind domain
Prime order of an ideal
Primitive element theorem
Principal ideal
Principal ideal domain
Principal ideal ring
Probability density function
Probability model
Probability theory MOC
Product topology
Products and coproducts
Projection operator
Projective code
Projective general linear group
Projective object
Projective system of a linear code
Proper class
Proper sublattice
Properly discontinuous group action
Proto-Central IE sample conjugations
Proto-Vṛddhi
Proving continuity with a subbasis
Proving open map with a subbasis
Pseudoautistic category
Publishing my PKM
Pushforward and pullback of an isomorphism
Pushforward measure
QM in 3D position-space
QM of a free particle in 1D
QM of a particle in a 1D Dirac delta potential
QM of a particle in a 1D harmonic oscillator
QM of a particle in a 1D infinite square well
QM of a particle in a 3D infinite square well
QM of particle in a 1D finite square well
Quadratic field
Quadratic form
Quadratic integers
Quadratic Lie algebra
Quantile function
Quantum teleportation
Quasi-perfect code
Quasistatic expression for work
Quasistatic process
Quiver homomorphism
Quiver representation
Quiver subrepresentation
Quotient algebra
Quotient group
Quotient Lie algebra
Quotient quadratic space
Quotient set
Quotient topology
R-monoid
R-semigroup
Radical of a Lie algebra
Radical of a quadratic space
Raising and lowering of indices
Random function
Random sample
Rank
Rank-nullity theorem
Rational lattice
Rationals adjoin sqrt(-14)
Rationals adjoin sqrt(-21)
Rationals adjoin sqrt(-3)
Real embedded manifold
Real projective space
Real quadratic field
Real random variable
Recursion
Reducibility of modules
Reducibility of quiver representations
Reducibility of representations
Reduction of order (homogenous second-order differential equation)
Regular epimorphism
Regular group action
Regular group representation
Regular measure
Regular monomorphism
Regular representation
Relation class
Relation set
Relationship between I𝑎𝑏I and I𝑎I𝑏I
Relationship between Kinetic Energy and Temperature
Relatively prime ideals
Representation of a finite group
Representation of an abelian group
Reverse order integration
Reverse triangle inequality
Ring
Ring homomorphism
Ring of integers of a number field
Rng
Rng homomorphism
Rolle's theorem
Row and column space
Russell's paradox
Russell's paradox for categories
Sample excess kurtosis
Sample moment
Sample skewness
Scalar field
Scalar field Lagrangian
Scalar transformation criterion
Schur's lemma
Second countable implies Lindelöf
Second law of thermodynamics
Seifert-Van Kampen-Brown theorem
Self-dual rational lattice
Semantic Breaks
Semi-Riemannian manifold
Semi-standard Young tableau
Semidirect product
Semidirect product of Lie algebras
Semilocal simple connectedness
Seminormed vector space
Semisimple module
Separability of a finite extension
Separable degree of an extension
Separable differential equation
Separable polynomial
Sequential closedness
Sequential continuity
Sequentially compact space
Set
Sheet number of a connected covering
Shifted graded module
Signature of a number field
Signature of a semi-Riemannian manifold
Simple connectedness
Simple Lie algebra
Simple module
Simple object
Simplicity of an algebraic extension
Simpson's lemma
Single faced string diagram
Single point space
Singular point
Skeletal categories are equivalent iff they are isomorphic
Skeletal category
Skeleton category
Skewness
Small set
Solvable Lie algebra
Solving non-homogenous second order ODEs
Span
Specific heat of an ideal gas
Specification Axiom Schema
Spectrum of a McKay graph
Spectrum of an algebraic element
Sphere packing condition for a perfect code
Sphere space
Split epimorphism
Split monomorphism
Split short exact sequence
Splitting of prime ideals in a number field
Square sum of irrep dimensions
Stabilizer group
Standard Heisenberg algebra for QM
Statistical moment
Strict category of graded vector spaces
Strict monoidal category
Subalgebra generated by an algebraic element
Subalgebra over a field
Subcategory
Subclass
Subfield
Subgroup
Submanifold
Submodule
Subring
Subrng
Subset
Subspace topology
Sum of commuting Lie algebra representations
Sum of independent random variables
Summable endomorphisms
Support of a map
Sure convergence
Surjectivity, injectivity, and bijectivity
Sweedler's large Hopf algebra
Sweedler's small Hopf algebra
Sylow's theorem
Symmetric algebra
Symmetric distribution
Symmetric group
Symmetric monoidal category
Symplectic group representation
System of linear equations
Systematic code
Table of Laplace transforms
Tag Glossary
Taylor's theorem
Tensor algebra
Tensor field
Tensor product of a Lie algebra and a commutative algebra
Tensor product of algebras
Tensor product of Lie algebra representations
Tensor product of modules over a commutative ring
Tensor product with a 1-dimensional representation
The characteristic of an integral domain is 0 or prime
The conjugate of an n-cycle is an n-cycle
The continuous image of a compact space is compact
The deck transformation group acts properly discontinuously
The group product of elements in commuting subgroups generate a subgroup
The image map of a bijection is a bijection
The image of a group homomorphism is a subgroup
The intersection of nested inhabited Hausdorff-compact sets is inhabited
The intersection of nested inhabited sets may be empty
The intersection of normal subgroups is a normal subgroup
The intersection of subgroups is a subgroup
The intersection of topologies on a fixed set is again a topology
The order of a cyclic group equals the order of its generator
The order of an element and its inverse are the same
The orthogonal complement of an invariant subspace under a unitary operator is invariant
The polynomial ring over an integral domain is an integral domain
The preïmage of the image and image of the preïmage are not necessarily the identity
The regular representation contains all irreducible representations
The relationship between temperature and heat in a solid or liquid
The ring of integers of a number field forms a lattice
Thermal equilibrium
Thermodynamic potential
Thin category
Thorn cluster
Tikhonov's theorem
Topological basis
Topological group
Topological manifold
Topological retraction
Topological space
Topological subbasis
Topological vector space
Torsion
Torsion group
Torsion subgroup of an abelian group
Torsion tensor
Torus topology
Total derivative
Totally ordered set
Trace form
Transition map
Transitive group action
Triangular Lie algebra
Triangular module
Trivial and alternating characters of a finite symmetric group in tensor product decomposition
Trivial category
Trivial group
Trivial irrep carrying ideal of the group ring
Trivial measure
Twisted affine Lie algebra
Uniform continuity
Uniform norm
Unique factorization of ideals
Unique group involutions are central
Unit
Unitary operator
Universal enveloping algebra
Universal Relation Axiom
Universality of the uniform distribution
UPMATH
Use folders for separating essentially different kinds of content.
Use MOCs for grouping closely related content with navigation in mind.
Use tags for organised, but loose and fluid "categories" of otherwise unrelated notes.
Using Folders, MOCs and Tags
Vacuum space
Vandermonde matrix
Vector basis
Vector field
Vector space
Vector space over a field extension
Vector subspace
Velocity selector for charged particles
Virasoro algebra
Weight enumerator
Yoneda embedding
Yoneda lemma
Young diagram
Young operator
Young tableau
Young's inequality
Zero morphism
Zero ring
Zeroth Law of Thermodynamics
ZF
ZFC
𝜇-estimator
σ-algebra

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